Guaranteed and computable error bounds for approximations constructed by an iterative decoupling of the Biot problem. (English) Zbl 1524.76458
Summary: The paper is concerned with guaranteed a posteriori error estimates for a class of evolutionary problems related to poroelastic media governed by the quasi-static linear Biot equations. The system is decoupled by employing the fixed-stress split scheme, which leads to an iteratively solved semi-discrete system. The error bounds are derived by combining a posteriori estimates for contractive mappings with functional type error control for elliptic partial differential equations. The estimates are applicable to any approximation in the admissible functional space and are independent of the discretization method. They are fully computable, do not contain mesh-dependent constants, and provide reliable global estimates of the error measured in the energy norm. Moreover, they suggest efficient error indicators for the distribution of local errors and can be used in adaptive procedures.
MSC:
| 76S05 | Flows in porous media; filtration; seepage |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
Biot problem; fixed-stress split iterative scheme; a posteriori error estimates; contraction mappingsReferences:
| [1] | Terzaghi, K., Principle of soil mechanics (1925), Eng. News Record, A Series of Articles |
| [2] | Biot, M., General theory of three-dimensional consolidation, J. Appl. Phys., 12, 155-164 (1941) · JFM 67.0837.01 |
| [3] | Biot, M., Theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys., 26, 182-185 (1955) · Zbl 0067.23603 |
| [4] | Coussy, O., Mechanics of Porous Continua (1995), Wiley: Wiley West Sussex · Zbl 0838.73001 |
| [5] | Showalter, R. E., Diffusion in poro-elastic media, J. Math. Anal. Appl., 251, 310-340 (2000) · Zbl 0979.74018 |
| [6] | Showalter, R. E.; Stefanelli, U., Diffusion in poro-plastic media, Math. Methods Appl. Sci., 27, 2131-2151 (2004) · Zbl 1095.74011 |
| [7] | Murad, M. A.; Loula, A. F.D., Improved accuracy in finite element analysis of Biot’s consolidation problem, Comput. Methods Appl. Mech. Engrg., 95, 359-382 (1992) · Zbl 0760.73068 |
| [8] | Mikelić, A.; Wheeler, M. F., Theory of the dynamic Biot-Allard equations and their link to the quasi-static Biot system, J. Math. Phys., 53, Article 123702 pp. (2012), 15 · Zbl 1331.35283 |
| [9] | Settari, A.; Mourits, F., A coupled reservoir and geomechanical simulation system, SPE J., 3, 219-226 (1998) |
| [10] | Almani, T.; Kumar, K.; Wheeler, M. F., Convergence and error analysis of fully discrete iterative coupling schemes for coupling flow with geomechanics, Comput. Geosci., 21, 1157-1172 (2017) · Zbl 1396.76047 |
| [11] | Kumar, K.; Almani, T.; Singh, G.; Wheeler, M. F., Multirate undrained splitting for coupled flow and geomechanics in porous media, (Numerical Mathematics and Advanced Applications—ENUMATH 2015. Numerical Mathematics and Advanced Applications—ENUMATH 2015, Lect. Notes Comput. Sci. Eng., vol. 112 (2016), Springer: Springer [Cham]), 431-440 · Zbl 1387.76100 |
| [12] | Almani, T.; Kumar, K.; Dogru, A.; Singh, G.; Wheeler, M. F., Convergence analysis of multirate fixed-stress split iterative schemes for coupling flow with geomechanics, Comput. Methods Appl. Mech. Engrg., 311, 180-207 (2016) · Zbl 1439.74183 |
| [13] | Vermeer, P. A.; Verruijt, A., An accuracy condition for consolidation by finite elements, Int. J. Numer. Anal. Methods Geomech., 5, 1-14 (1981) · Zbl 0456.73060 |
| [14] | Reed, M. B., An investigation of numerical errors in the analysis of consolidation by finite elements, Int. J. Numer. Anal. Methods Geomech., 8, 243-257 (1984) · Zbl 0536.73089 |
| [15] | Zienkiewicz, O. C.; Shiomi, T., Dynamic behaviour of saturated porous media; the generalized Biot formulation and its numerical solution, Int. J. Numer. Anal. Methods Geomech., 8, 71-96 (1984) · Zbl 0526.73099 |
| [16] | Murad, M. A.; Loula, A. F.D., On stability and convergence of finite element approximations of Biot’s consolidation problem, Internat. J. Numer. Methods Engrg., 37, 645-667 (1994) · Zbl 0791.76047 |
| [17] | Murad, M. A.; Thomée, V.; Loula, A. F.D., Asymptotic behavior of semidiscrete finite-element approximations of Biot’s consolidation problem, SIAM J. Numer. Anal., 33, 1065-1083 (1996) · Zbl 0854.76053 |
| [18] | Korsawe, J.; Starke, G., A least-squares mixed finite element method for Biot’s consolidation problem in porous media, SIAM J. Numer. Anal., 43, 318-339 (2005) · Zbl 1086.76041 |
| [19] | Nordbotten, J. M., Stable cell-centered finite volume discretization for Biot equations, SIAM J. Numer. Anal., 54, 942-968 (2016) · Zbl 1382.76187 |
| [20] | Chen, Y.; Luo, Y.; Feng, M., Analysis of a discontinuous Galerkin method for the Biot’s consolidation problem, Appl. Math. Comput., 219, 9043-9056 (2013) · Zbl 1290.74038 |
| [21] | Boffi, D.; Botti, M.; Di Pietro, D. A., A nonconforming high-order method for the Biot problem on general meshes, SIAM J. Sci. Comput., 38, A1508-A1537 (2016) · Zbl 1337.76042 |
| [22] | Vignollet, J.; May, S.; de Borst, R., On the numerical integration of isogeometric interface elements, Internat. J. Numer. Methods Engrg., 102, 1733-1749 (2015) · Zbl 1352.65087 |
| [23] | Chan, S.-H.; Phoon, K.-K.; Lee, F. H., A modified jacobi preconditioner for solving ill-conditioned Biots consolidation equations using symmetric quasi-minimal residual method, Int. J. Numer. Anal. Methods Geomech., 25, 1001-1025 (2001) · Zbl 1065.74605 |
| [24] | Phoon, K. K.; Toh, K. C.; Chan, S. H.; Lee, F. H., An efficient diagonal preconditioner for finite element solution of Biot consolidation equations, J. Numer. Methods Engrg., 55, 377-400 (2002) · Zbl 1076.74558 |
| [25] | White, J. A.; Borja, R. I., Block-preconditioned Newton-Krylov solvers for fully coupled flow and geomechanics, Comput. Geosci., 15, 647-659 (2011) · Zbl 1367.76034 |
| [26] | Haga, J. B.; Osnes, H.; Langtangen, H. P., A parallel block preconditioner for large-scale poroelasticity with highly heterogeneous material parameters, Comput. Geosci., 16, 723-734 (2012) |
| [27] | Axelsson, O.; Blaheta, R.; Byczanski, P., Stable discretization of poroelasticity problems and efficient preconditioners for arising saddle point type matrices, Comput. Vis. Sci., 15, 191-207 (2012) · Zbl 1388.74035 |
| [28] | Haga, J. B.; Osnes, H.; Langtangen, H. P., A parallel block preconditioner for large-scale poroelasticity with highly heterogeneous material parameters, Comput. Geosci., 16, 723-734 (2012) |
| [29] | Castelletto, N.; White, J.; Tchelepi, H., Accuracy and convergence properties of the fixed-stress iterative solution of two-way coupled poromechanics, Int. J. Numer. Anal. Methods Geomech., 39, 1593-1618 (2015) |
| [30] | Castelletto, N.; White, J.; Ferronato, M., Scalable algorithms for three-field mixed finite element coupled poromechanics, J. Comput. Phys., 327, 894-918 (2016) · Zbl 1373.76312 |
| [31] | Rhebergen, S.; Wells, G. N.; Katz, R. F.; Wathen, A. J., Analysis of block preconditioners for models of coupled magma/mantle dynamics, SIAM J. Sci. Comput., 36, A1960-A1977 (2014) · Zbl 1299.86001 |
| [32] | Rhebergen, S.; Wells, G. N.; Wathen, A. J.; Katz, R. F., Three-field block preconditioners for models of coupled magma/mantle dynamics, SIAM J. Sci. Comput., 37, A2270-A2294 (2015) · Zbl 1327.65055 |
| [33] | Hong, Q.; Kraus, J., Uniformly stable discontinuous Galerkin discretization and robust iterative solution methods for the Brinkman problem, SIAM J. Numer. Anal., 54, 2750-2774 (2016) · Zbl 1346.76068 |
| [34] | Lee, J. J.; Mardal, K.-A.; Winther, R., Parameter-robust discretization and preconditioning of Biot’s consolidation model, SIAM J. Sci. Comput., 39, A1-A24 (2017) · Zbl 1381.76183 |
| [35] | Bærland, T.; Lee, J. J.; Mardal, K.-A.; Winther, R., Weakly imposed symmetry and robust preconditioners for Biot’s consolidation model, Comput. Methods Appl. Math., 17, 377-396 (2017) · Zbl 1421.74095 |
| [36] | D. Allen, Stability, accuracy, and efficiency of sequential methods for coupled flow and geomechanics, Technical Report SPE119084, The SPE Reservoir Simulation Symposium, Houston, Texas, 2009. |
| [37] | White, J. A.; Castelletto, N.; Tchelepi, H. A., Block-partitioned solvers for coupled poromechanics: a unified framework, Comput. Methods Appl. Mech. Engrg., 303, 55-74 (2016) · Zbl 1425.74497 |
| [38] | Gaspar, F. J.; Rodrigo, C., On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics, Comput. Methods Appl. Mech. Engrg., 326, 526-540 (2017) · Zbl 1439.74413 |
| [39] | Hong, Q.; Kraus, J., Parameter-robust stability of classical three-field formulation of Biot consolidation model, 202-226 (2018) · Zbl 1398.65046 |
| [40] | Meunier, S., Analyse d’erreur a posteriori pour les couplages hydro-Mécaniques et mise en oeuvre dans (2007), École des Ponts ParisTech, (Ph.D. thesis) |
| [41] | Ern, A.; Meunier, S., A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems, M2AN Math. Model. Numer. Anal., 43, 353-375 (2009) · Zbl 1166.76036 |
| [42] | Vohralík, M.; Wheeler, M. F., A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows, Comput. Geosci., 17, 789-812 (2013) · Zbl 1393.76069 |
| [43] | Yousef, S., A Posteriori Error Estimates and Adaptivity Based on Stopping Criteria and Adaptive Mesh Re Finement for Multiphase and Thermal Flows. Application to Steam-Assisted Gravity Drainage (2013), Université Pierre et Marie Curie: Université Pierre et Marie Curie Paris VI, (Ph.D. thesis) |
| [44] | Di Pietro, D. A.; Vohralík, M.; Yousef, S., An a posteriori-based, fully adaptive algorithm with adaptive stopping criteria and mesh refinement for thermal multiphase compositional flows in porous media, Comput. Math. Appl., 68, 2331-2347 (2014) · Zbl 1362.65097 |
| [45] | Di Pietro, D. A.; Flauraud, E.; Vohralík, M.; Yousef, S., A posteriori error estimates, stopping criteria, and adaptivity for multiphase compositional Darcy flows in porous media, J. Comput. Phys., 276, 163-187 (2014) · Zbl 1349.76323 |
| [46] | Ahmed, E.; Hassan, S. A.; Japhet, C.; Kern, M.; Vohralík, M., A Posteriori Error Estimates and Stopping Criteria for Space-Time Domain Decomposition for Two-Phase Flow Between Different Rock TypesTechnical Report hal-01540956, Version 1 (2017), HAL |
| [47] | Larsson, F.; Runesson, K., A sequential-adaptive strategy in space-time with application to consolidation of porous media, Comput. Methods Appl. Mech. Engrg., 288, 146-171 (2015) · Zbl 1423.76252 |
| [48] | Riedlbeck, R.; Di Pietro, D. A.; Ern, A.; Granet, S.; Kazymyrenko, K., Stress and flux reconstruction in Biot’s poro-elasticity problem with application to a posteriori error analysis, Comput. Math. Appl., 73, 1593-1610 (2017) · Zbl 1370.76086 |
| [49] | Bertrand, F.; Moldenhauer, M.; Starke, G., A Posteriori Error Estimation for Planar Linear Elasticity by Stress Reconstruction. A Posteriori Error Estimation for Planar Linear Elasticity by Stress Reconstruction, Comput. Methods Appl. Math., 19, 3, 663-679 (2019) · Zbl 1420.65117 |
| [50] | Ahmed, E.; Radu, F. A.; Nordbotten, J. M., Adaptive poromechanics computations based on a posteriori error estimates for fully mixed formulations of Biot’s consolidation model, Comput. Methods Appl. Mech. Engrg., 347, 264-294 (2019) · Zbl 1440.74117 |
| [51] | Ahmed, E.; Nordbotten, J. M.; Radu, F. A., Adaptive asynchronous time-stepping, stopping criteria, and a posteriori error estimates for fixed-stress iterative schemes for coupled poromechanics problems, J. Comput. Appl. Math., 364 (2020) · Zbl 1524.76439 |
| [52] | Nordbotten, J. M.; Rahman, T.; Repin, S. I.; Valdman, J., A posteriori error estimates for approximate solutions of the Barenblatt-Biot poroelastic model, Comput. Methods Appl. Math., 10, 302-314 (2010) · Zbl 1283.65100 |
| [53] | Banach, S., Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3, 133-181 (1922) · JFM 48.0201.01 |
| [54] | Repin, S., (A Posteriori Estimates for Partial Differential Equations. A Posteriori Estimates for Partial Differential Equations, Radon Series on Computational and Applied Mathematics, vol. 4 (2008), Walter de Gruyter GmbH & Co. KG: Walter de Gruyter GmbH & Co. KG Berlin) · Zbl 1162.65001 |
| [55] | Ženíšek, A., The existence and uniqueness theorem in Biot’s consolidation theory, Apl. Mat., 29, 194-211 (1984) · Zbl 0557.35005 |
| [56] | Showalter, R. E.; Su, N., Partially saturated flow in a poroelastic medium, Discrete Contin. Dyn. Syst. Ser. B, 1, 403-420 (2001) · Zbl 1004.76090 |
| [57] | Mikelić, A.; Wheeler, M. F., Convergence of iterative coupling for coupled flow and geomechanics, Comput. Geosci., 17, 455-461 (2013) · Zbl 1392.35235 |
| [58] | Mikelić, A.; Wang, B.; Wheeler, M. F., Numerical convergence study of iterative coupling for coupled flow and geomechanics, Comput. Geosci., 18, 325-341 (2014) · Zbl 1386.76115 |
| [59] | Kim, J.; Tchelepi, H. A.; Juanes, R., Stability and convergence of sequential methods for coupled flow and geomechanics: drained and undrained splits, Comput. Methods Appl. Mech. Engrg., 200, 2094-2116 (2011) · Zbl 1228.74106 |
| [60] | Kim, J.; Tchelepi, H. A.; Juanes, R., Stability and convergence of sequential methods for coupled flow and geomechanics: fixed-stress and fixed-strain splits, Comput. Methods Appl. Mech. Engrg., 200, 1591-1606 (2011) · Zbl 1228.74101 |
| [61] | Repin, S., A posteriori estimates for approximate solutions of variational problems with strongly convex functionals, Probl. Math. Anal., 17, 199-226 (1997) · Zbl 0941.65059 |
| [62] | Repin, S., A posteriori error estimation for variational problems with uniformly convex functionals, Math. Comp., 69, 481-500 (2000) · Zbl 0949.65070 |
| [63] | Neittaanmäki, P.; Repin, S., (Reliable Methods for Computer Simulation. Reliable Methods for Computer Simulation, Studies in Mathematics and its Applications, vol. 33 (2004), Elsevier Science B.V.: Elsevier Science B.V. Amsterdam), Error control and a posteriori estimates · Zbl 1076.65093 |
| [64] | Both, J. W.; Borregales, M.; Nordbotten, J. M.; Kumar, K.; Radu, F. A., Robust fixed stress splitting for Biot’s equations in heterogeneous media, Appl. Math. Lett., 68, 101-108 (2017) · Zbl 1383.74025 |
| [65] | Both, J. W.; Koecher, U., Numerical investigation on the fixed-stress splitting scheme for Biot’s equations: Optimality of the tuning parameter, Numerical mathematics and advanced applications—ENUMATH 2017, 789-797 (2019) · Zbl 1425.76132 |
| [66] | Repin, S.; Sauter, S., Functional a posteriori estimates for the reaction-diffusion problem, C. R. Acad. Sci., Paris, 343, 349-354 (2006) · Zbl 1100.65093 |
| [67] | Arnold, D. N.; Brezzi, F., Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér., 19, 7-32 (1985) · Zbl 0567.65078 |
| [68] | Arnold, D. N., Mixed finite element methods for elliptic problems, Comput. Methods Appl. Mech. Engrg., 82, 281-300 (1990), Reliability in computational mechanics (Austin, TX, 1989) · Zbl 0729.73198 |
| [69] | John, V., A numerical study of a posteriori error estimators for convection-diffusion equations, Comput. Methods Appl. Mech. Engrg., 190, 757-781 (2000) · Zbl 0973.76049 |
| [70] | Girault, V.; Wheeler, M. F.; Ganis, B.; Mear, M. E., A lubrication fracture model in a poro-elastic medium, Math. Models Methods Appl. Sci., 25, 587-645 (2015) · Zbl 1309.76194 |
| [71] | Ladyzhenskaya, O. A., The Boundary Value Problems of Mathematical Physics (1985), Springer: Springer New York · Zbl 0588.35003 |
| [72] | Ladyzhenskaya, O. A.; Solonnikov, V. A.; Uraltseva, N., Linear and Quasilinear Equations of Parabolic Type (1967), Nauka: Nauka Moscow · Zbl 0164.12302 |
| [73] | Zeidler, E., Nonlinear Functional Analysis and Its Applications. II/A (1990), Springer-Verlag: Springer-Verlag New York, Linear monotone operators, Translated from the German by the author and Leo F. Boron · Zbl 0684.47029 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
